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Let $X$ be a compact space (e.g. the Cantor Set) and let $T:X\to X$ be a minimal homeomorphism (meaning that every orbit is dense in $X$).

If $U\subseteq X$ is an open subset (clopen if X is the Cantor set), let $T_U:U\to U$ be the first return map i.e., an element $x\in U$ is sent to its first return point in $U$. (Such a point exists by the minimality of $T$). Easy to see that $(U,T_U)$ is another minimal system.

Specific question:

A paper I am reading (link) states (without proof) that if $(X,T)$ is not (conjugate to) an odemeter then so is $(U,T_U)$. I was unable to observe this.

General question:

In general, what relations do the systems $(X,T)$ and $(U,T_U)$ have?

Thank you in advance.

Mustafa Gokhan Benli
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  • It might be good to add the paper you are reading. As a hint, one can reconstruct $T$ from the induced system $T_U$. – Alp Uzman Jul 31 '23 at 17:03
  • Added link to article – Mustafa Gokhan Benli Aug 01 '23 at 11:14
  • @AlpUzman can you elobarete please? – Mustafa Gokhan Benli Aug 02 '23 at 11:13
  • To build some intuition, the graph https://www.desmos.com/calculator/v74aykkywa might be useful. What I am getting at is this: call $\rho=\rho_{T,U}$ the first return time function to $U$ under $T$, so that $T_U(x)=T^{\rho_{T,U}(x)}(x)$. Then considering $\rho$ as a roof function over $T_U$, the suspension system is isomorphic to the original $T$. This in particular rearranges $X$ as a tower of $U$. Once you have this it should be fairly straightforward to see that the (integral) suspension of any odometer ought to be an odometer. – Alp Uzman Aug 03 '23 at 13:42
  • I would also highly recommend reediting your post to attract more potential helping hands. For instance you don't define the crucial definitions that a user who might be helpful may not know. Further your link is behind a paywall, or it requires access to Mathscinet, whereas there is a publicly available preprint. It also makes a post more long lasting if instead of a terse link more explicit reference information is given, e.g. the author and the title or article (however unlikely it is for Mathscinet to go down any time soon). – Alp Uzman Aug 03 '23 at 13:49

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